Problem 59 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} x^{2}+y^{2}=169 \\ x^{2}-8 y=104 \end{array}\right.$$

Step-by-Step Solution

Answer

The exact solution depends on the specific approximations obtained from the graphing tool and their verification with the system. After graphing and verification, two points of intersection should be found, and their coordinates should satisfy both equations of the system.

Step 1: Graph the given equations

Start by graphing the given equations. Using a graphing tool such as a graphing calculator or software like Desmos, GeoGebra or similar, graph both equations \(x^{2}+y^{2}=169\) and \(x^{2}-8y=104\) on the same graph. The graph of \(x^{2}+y^{2}=169\) is a circle with radius 13 centered at the origin. The graph of \(x^{2}-8y=104\) is a parabola that opens upward.

Step 2: Approximate Intersection Points

Your next task is to identify and approximate the points where the parabola crosses the circle. You should be able to see two points of intersection. Using cursor or mouse controls provided in the graphing tool, find the points of intersection. Round your results to three decimal places.

Step 3: Verify the solutions

Now, you have to verify your answers by plugging the x and y coordinates of the intersections back into the original equations (system). They should satisfy both the equations. Remember to calculate for both intersection points.

Step 4: Put the results together

Finally, compile the points of intersection and make sure they are rounded to three decimal places. Ensure they satisfy both equations of the system. The points of intersection are your solutions.

Key Concepts

Equations System

An equations system comprises two or more different equations that share common variables. Solving the system means finding the values of the variables that satisfy all equations simultaneously. In our exercise, the system includes the following equations:

• The first equation is a circle equation: \(x^{2}+y^{2}=169\)
• The second equation is a parabola equation: \(x^{2}-8y=104\)

To solve the system on a graphing utility, one must graph both equations on the same set of axes and find where they intersect. The solutions to the system are the coordinates of these intersection points. Each intersection point must satisfy both equations when substituted back in, effectively making them the solution of the system.

Intersection Points

Finding the intersection points of two graphs is crucial when solving a system of equations graphically. In simpler terms, intersection points are the specific coordinates where two different graphs meet or cross each other.
Using a graphing tool simplifies this process considerably:

• First, plot both graphs accurately.
• Then, use the software’s tool (like a cursor or marker) to locate the precise spots where the graphs intersect.

Round these intersection points to three decimal places for accuracy. In the context of our specific problem, these are the points where the circle and the parabola touch each other. These coordinates should be verified by substituting back into the original equations to ensure they satisfy both.

Circle Equation

The equation of a circle is a way to describe a circle in mathematical terms. The standard form most commonly used is:
\[(x - h)^2 + (y - k)^2 = r^2\]
where

• \(h,k\) are the coordinates of the center of the circle
• \(r\) is the radius of the circle.

In our exercise, the circle equation is a variation simplified by being centered at the origin \((0,0)\), and is represented as:
\(x^{2}+y^{2}=169\)
Here, the center is at the origin, and the radius is 13, derived from \(r^2 = 169\). Graphing this equation will result in a perfect circle on a coordinate plane, making it easier to spot intersections with other graphs.

Parabola Equation

A parabola is represented by a quadratic equation and can open upwards, downwards, left, or right, depending on the form and coefficients of the equation. The general form of a parabola equation is:
\(y = ax^2 + bx + c\)
In this exercise, the parabola equation given is:
\(x^{2} - 8y = 104\)
By rearranging it into a more standard form, relying on \(y\), we get it as:
\(y = \frac{1}{8}x^{2} - 13\)
This equation indicates a parabola that opens upwards, with the vertex below the x-axis due to the \(-13\) adjustment. The accurate plotting of this parabola on the same graph as the circle is crucial to find out where these two shapes intersect, which provides the intersection points vital to solving the system of equations.